Multiple description transform coding (MDTC) is a type of joint source-channel coding (JSC) designed for transmission channels which are subject to failure or "erasure." The objective of MDTC is to ensure that a decoder which receives an arbitrary subset of the channels can produce a useful reconstruction of the original signal. One type of MDTC introduces correlation between transmitted coefficients in a known, controlled manner so that lost coefficients can be statistically estimated from received coefficients. This correlation is used at the decoder at the coefficient level, as opposed to the bit level, so it is fundamentally different than techniques that use information about the transmitted data to produce likelihood information for the channel decoder. The latter is a common element in other types of JSC coding systems, as shown, for example, in P. G. Sherwood and K. Zeger, "Error Protection of Wavelet Coded Images Using Residual Source Redundancy," Proc. of the 31.sup.st Asilomar Conference on Signals, Systems and Computers, November 1997. Other types of MDTC may be based on techniques such as frame expansions, as described in V. K. Goyal et al., "Multiple Description Transform Coding: Robustness to Erasures Using Tight Frame Expansions," In Proc. IEEE Int. Symp. Inform. Theory, August 1998.
A known MDTC technique for coding pairs of independent Gaussian random variables is described in M. T. Orchard et al., "Redundancy Rate-Distortion Analysis of Multiple Description Coding Using Pairwise Correlating Transforms," Proc. IEEE Int. Conf. Image Proc., Santa Barbara, Calif., October 1997. This MDTC technique provides optimal 2.times.2 transforms for coding pairs of signals for transmission over two channels. However, this technique as well as other conventional techniques fail to provide optimal generalized n.times.m transforms for coding any n signal components for transmission over any m channels. In addition, conventional transforms such as those in the M. T. Orchard et al. reference fail to provide a sufficient number of degrees of freedom, and are therefore unduly limited in terms of design flexibility. Moreover, the optimality of the 2.times.2 transforms in the M. T. Orchard et al. reference requires that the channel failures be independent and have equal probabilities. The conventional techniques thus generally do not provide optimal transforms for applications in which, for example, channel failures either are dependent or have unequal probabilities, or both. These and other drawbacks of conventional MDTC prevent its effective implementation in many important applications.